Biomolecular processes are typically modeled using chemical reaction networks coupled to infinitely large chemical reservoirs. A difference in chemical potential between these reservoirs can drive the system into a non-equilibrium steady state (NESS). In reality, these processes take place in finite systems containing a finite number of molecules. In such systems, a NESS can be reached with the help of an externally driven pump for which we introduce a simple model. Crucial parameters are the pumping rate and the finite size of the chemical reservoir. We apply this model to a simple biochemical oscillator, the Brusselator, and quantify the performance using the number of coherent oscillations. As a surprising result, we find that higher precision can be achieved with finite-size reservoirs even though the corresponding current fluctuations are larger than in the ideal infinite case.
We develop a consistent stochastic thermodynamics for environments composed of thermodynamic reservoirs in an external conservative force field, that is environments described by the Generalized or Gibbs canonical ensemble. We demonstrate that small systems weakly coupled to such reservoirs exchange both heat and work by verifying a local detailed balance relation for the induced stochastic dynamics. Based on this analysis, we help to rationalize the observation that nonthermal reservoirs can increase the efficiency of thermodynamic heat engines.
We set up a rigorous thermodynamic description of reaction-diffusion systems driven out of equilibrium by time-dependent space-distributed chemostats. Building on the assumption of local equilibrium, nonequilibrium thermodynamic potentials are constructed exploiting the symmetries of the chemical network topology. It is shown that the canonical (resp. semigrand canonical) nonequilibrium free energy works as a Lyapunov function in the relaxation to equilibrium of a closed (resp. open) system and its variation provides the minimum amount of work needed to manipulate the species concentrations. T…
We reformulate stochastic thermodynamics in terms of noise realizations for Langevin systems in contact with multiple reservoirs and investigated the structure of the second laws of thermodynamics. We derive a hierarchy of fluctuation theorems when one degree of freedom of the system is affected by multiple reservoirs simultaneously, that is, when noise mixing occurs. These theorems and the associated second laws of thermodynamics put stricter bounds on the thermodynamics of Langevin systems. We apply our results to a stochastic machine in noise-mixing environments and demonstrate that our new…
We generalize stochastic thermodynamics to include information reservoirs. Such information reservoirs, which can be modeled as a sequence of bits, modify the second law. For example, work extraction from a system in contact with a single heat bath becomes possible if the system also interacts with an information reservoir. We obtain an inequality, and the corresponding fluctuation theorem, generalizing the standard entropy production of stochastic thermodynamics. From this inequality we can derive an information processing entropy production, which gives the second law in the presence of info…
We derive a linear thermodynamics theory for general Markov dynamics with both steady-state and time-periodic drivings. Expressions for thermodynamic quantities, such as mechanical and chemical work, heat and entropy production are obtained in terms of equilibrium probability distribution and the drivings. The entropy production is derived as a bilinear function of thermodynamic forces and the associated fluxes. We derive explicit formulae for the Onsager coefficients and use them to verify the Onsager-Casimir reciprocal relations. Our results are illustrated on a periodically driven quantum d…
After establishing stochastic thermodynamics for underdamped Langevin systems in contact with multiple reservoirs, we derive its overdamped limit using timescale separation techniques. The overdamped theory is different from the naive theory that one obtains when starting from overdamped Langevin or Fokker-Planck dynamics and only coincide with it in presence of a single reservoir. The reason is that the coarse-grained fast momenta dynamics reaches a nonequilibrium state which conducts heat in presence of multiple reservoirs. The underdamped and overdamped theory are both shown to satisfy fund…
Chemical waves constitute a known class of dissipative structures emerging in reaction-diffusion systems. They play a crucial role in biology, spreading information rapidly to synchronize and coordinate biological events. We develop a rigorous thermodynamic theory of reaction-diffusion systems to characterize chemical waves. Our main result is the definition of the proper thermodynamic potential of the local dynamics as a nonequilibrium free energy density and establishing its balance equation. This enables us to identify the dynamics of the free energy, of the dissipation, and of the work spe…
We show that non-Markovian effects of the reservoirs can be used as a resource to extract work from an Otto cycle. The state transformation under non-Markovian dynamics is achieved via a two-step process, namely an isothermal process using a Markovian reservoir followed by an adiabatic process. From second law of thermodynamics, we show that the maximum amount of extractable work from the state prepared under the non-Markovian dynamics quantifies a lower bound of non-Markovianity. We illustrate our ideas with an explicit example of non-Markovian evolution.
We analyze the entropy production and the maximal extractable work from a squeezed thermal reservoir. The nonequilibrium quantum nature of the reservoir induces an entropy transfer with a coherent contribution while modifying its thermal part, allowing work extraction from a single reservoir, as well as great improvements in power and efficiency for quantum heat engines. Introducing a modified quantum Otto cycle, our approach fully characterizes operational regimes forbidden in the standard case, such as refrigeration and work extraction at the same time, accompanied by efficiencies equal to u…
We obtain generalizations of the Kelvin-Planck, Clausius, and Carnot statements of the second law of thermodynamics, for situations involving information processing. To this end, we consider an information reservoir (representing, e.g. a memory device) alongside the heat and work reservoirs that appear in traditional thermodynamic analyses. We derive our results within an inclusive framework in which all participating elements -- the system or device of interest, together with the heat, work and information reservoirs -- are modeled explicitly by a time-independent, classical Hamiltonian. We p…
We study the non-equilibrium thermodynamics of a heat engine operating between two finite-sized reservoirs with well-defined temperatures. Within the linear response regime, it is found that the uniform temperature of the two reservoirs at final time $τ$ is bounded from below by the entropy production $σ_{\mathrm{min}}\propto1/τ$. We discover a general power-efficiency trade-off depending on the ratio of heat capacities ($γ$) of the reservoirs for the engine. And a universal efficiency at maximum average power of the engine for arbitrary $γ$ is obtained. For practical purposes, the operation p…
We consider macroscopic systems in weak contact with boundary reservoirs and under the action of external fields. We present an explicit formula for the Hamiltonian of such systems, from which we deduce the equation of motions, the action functional, the hydrodynamic equation for the adjoint dynamics, and a formula for the quasi-potential. We examine the case in which the external forcing depends on time and drives the system from one nonequilibrium state to another. We extend the results presented in [6] on thermodynamic transformations for systems in strong contact with boundary reservoirs t…
The thermodynamics of the Curzon-Ahlborn engine, which is a prototype of endoreversible engines, is elucidated. In particular, their criterion for adiabatic equilibration is revised. The so-called irreversibility of endoreversible engines arises from the selection of the coldest reservoir for heat rejection. Rather, if the reservoirs are allowed to come into thermal and mechanical contact, a mean value results which optimizes the work output and heat uptake, and is entirely reversible. The Carnot efficiency cannot be beaten because nothing is as cold as the coldest reservoir.
Spontaneous pattern formation is a fundamental scientific problem that has received much attention since the seminal theoretical work of Turing on reaction-diffusion systems. In molecular biophysics, this phenomena often takes place under the influence of large fluctuations. It is then natural to inquire about the precision of such pattern. In particular, spontaneous pattern formation is a nonequilibrium phenomenon, and the relation between the precision of a pattern and the thermodynamic cost associated with it remains unexplored. Here, we analyze this relation with a paradigmatic stochastic …
Redfield theory provides a closed kinetic description of a quantum system in weak contact with a very dense reservoir. Landau-Zener theory does the same for a time-dependent driven system in contact with a sparse reservoir. Using a simple model, we analyze the validity of these two theories by comparing their predictions with exact numerical results. We show that despite their a priori different range of validity, these two descriptions can give rise to an identical quantum master equation. Both theories can be used for a nonequilibrium thermodynamic description which we show is consistent wit…
In the past decade several theoretical Maxwell's demon models have been proposed exhibiting effects such as refrigerating, doing work at the cost of information, and some experiments have been done to realise these effects. Here we propose a model with a two level demon, information represented by a sequence of bits, and two heat reservoirs. Which reservoir the demon interact with depends on the bit. If information is pure, one reservoir will be refrigerated, on the other hand, information can be erased if temperature difference is large. Genuine examples of such a system are discussed.
We present an overview of the reaction coordinate approach to handling strong system-reservoir interactions in quantum thermodynamics. This technique is based on incorporating a collective degree of freedom of the reservoir (the reaction coordinate) into an enlarged system Hamiltonian (the supersystem), which is then treated explicitly. The remaining residual reservoir degrees of freedom are traced out in the usual perturbative manner. The resulting description accurately accounts for strong system-reservoir coupling and/or non-Markovian effects over a wide range of parameters, including regim…
We explore the effects of quantum mechanical squeezing on the nonequilibrium fluctuations of bosonic transport between two squeezed harmonic reservoirs and a two level system. A standard full counting statistics technique based on a quantum master equation is employed. We derive a nonzero thermodynamic affinity under equal temperature setting of the two squeezed reservoirs. The odd cumulants are shown to be independent of squeezing under symmetric conditions, whereas the even cumulants depend nonlinearly on the squeezing parameters. The odd and even cumulants saturate at two different but uniq…
Despite its historical importance, a perfect gas enclosed by a pistons and in contact with a thermal reservoirs is a system still largely under study. Its thermodynamic properties are not yet well understood when driven under non-equilibrium conditions. In particular, analytic formulas that describe the heat exchanged with the reservoir are rare. In this paper we prove a power series expansions for the heat when both the external force and the reservoir temperature are slowly varying over time but the overall process is not quasi-static. To do so, we use the dynamical equations from [Cerino \e…
Phenomenological nonequilibrium thermodynamics describes how fluxes of conserved quantities such as matter, energy and charge flow from outer reservoirs across a system, and how they irreversibly degrade from one form to another. Stochastic thermodynamics is formulated in terms of probability fluxes circulating in the system's configuration space. The consistency of the two frameworks is granted by the condition of local detailed balance, which specifies the amount of physical quantities exchanged with the reservoirs during single transitions between configurations. We demonstrate that the top…
There have been two distinct formalisms of thermodynamics of information: One is the measurement-feedback formalism, which concerns bipartite systems with measurement and feedback processes, and the other is the information reservoir formalism, which considers bit sequences as a thermodynamic fuel. In this paper, we derive a second-law-like inequality by applying the measurement-feedback formalism to information reservoirs, which provides a stronger bound of extractable work than any other known inequality in the same setup. In addition, we demonstrate that the Mandal-Jarzynski model, which is…
The phenomenon described by our title should surprise no one. What may be surprising though is how easy it is to produce a quantum system with this feature; moreover, that system is one that is often used for the purpose of showing how systems equilibrate. The violation can be variously manifested. In our detailed example, bringing a detuned 2-level system into contact with a monochromatic reservoir does not cause it to relax to the reservoir temperature; rather, the system acquires the reservoir's level-occupation-ratio.
We show that systems driven by an external force and described by Nose-Hoover dynamics allow for a consistent nonequilibrium thermodynamics description when the thermostatted variable is initially assumed in a state of canonical equilibrium. By treating the "real" variables as the system and the thermostatted variable as the reservoir, we establish the first and second law of thermodynamics. As for Hamiltonian systems, the entropy production can be expressed as a relative entropy measuring the system-reservoir correlations established during the dynamics.
We establish the foundations of a nonequilibrium theory of quantum thermodynamics for noninteracting open quantum systems strongly coupled to their reservoirs within the framework of the nonequilibrium Green functions (NEGF). The energy of the system and its coupling to the reservoirs are controlled by a slow external time-dependent force treated to first order beyond the quasistatic limit. We derive the four basic laws of thermodynamics and characterize reversible transformations. Stochastic thermodynamics is recovered in the weak coupling limit.
In a recent perspective article, Horowitz and Gingrich discuss thermodynamic uncertainty relations that have been derived using "stochastic thermodynamics", a theory based on a hypothesis known as local detailed balance. The authors examined the foundations of this theory in their "Box 1: A brief primer on local detailed balance and stochastic thermodynamics", where a kinetic scheme for transport of particles between two reservoirs is presented. Horowitz and Gingrich arrive at a relationship between the probability to cycle through the states in one order vs. the probability to cycle in the re…
We test Boltzmann's H-theorem for several models of particle random walk. We study the influence of interaction between the particle and reservoir/detectors on entropy and find entropy increasing in time for some models and behaving non-monotonically for others. The key mechanism affecting the entropy growth is the quantum entanglement between the system and the reservoir. We discuss the details of the system-reservoir interaction, such as presence of the interference in the system and number of interactions with detector parts, and their impact on the monotonicity of entropy.
The hallmark of active matter is the autonomous directed motion of its microscopic constituents driven by consumption of energy resources. This leads to the emergence of large scale dynamics and structures without any equilibrium equivalent. Though active field theories offer a useful hydrodynamic description, it is unclear how to properly quantify the energetic cost of the dynamics from such a coarse-grained description. We provide a thermodynamically consistent framework to identify the energy exchanges between active systems and their surrounding thermostat at the hydrodynamic level. Based …
The simulation of open molecular systems requires explicit or implicit reservoirs of energy and particles. Whereas full atomistic resolution is desired in the region of interest, there is some freedom in the implementation of the reservoirs. Here, we construct a combined, explicit reservoir by interfacing the atomistic region with regions of point-like, non-interacting particles (tracers) embedded in a thermodynamic mean field. The tracer molecules acquire atomistic resolution upon entering the atomistic region and equilibrate with this environment, while atomistic molecules become tracers gov…
We consider open quantum systems weakly coupled to thermal reservoirs and subjected to quantum feedback operations triggered with or without delay by monitored quantum jumps. We establish a thermodynamic description of such system and analyze how the first and second law of thermodynamics are modified by the feedback. We apply our formalism to study the efficiency of a qubit subjected to a quantum feedback control and operating as a heat pump between two reservoirs. We also demonstrate that quantum feedbacks can be used to stabilize coherences in nonequilibrium stationary states which in some …
We verify the finite time fluctuation theorem for a linear Ising chain at its ends in contact with heat reservoirs. Analytic results are derived for a chain consisting of only two spins. The system can be mapped onto a model for particle transport, namely the symmetric exclusion process, in contact with thermal and particle reservoirs. We modify the symmetric exclusion process to represent a thermal engine and reproduce universal features of the efficiency at maximum power.
A general thermodynamic framework is presented for open quantum systems in fixed contact with a thermal reservoir. The first and second law are obtained for arbitrary system-reservoir coupling strengths, and including both factorized and correlated initial conditions. The thermodynamic properties are adapted to the generally strong coupling regime, approaching the ones defined for equilibrium, and their standard weak-coupling counterparts as appropriate limits. Moreover, they can be inferred from measurements involving only system observables. Finally, a thermodynamic signature of non-Markovia…
Thermodynamics with multiple-conserved quantities offers a promising direction for designing novel devices. For example, Vaccaro and Barnett's [J. A. Vaccaro and S. M. Barnett, Proc. R. Soc. A 467, 1770 (2011); S. M. Barnett and J. A. Vaccaro, Entropy 15, 4956 (2013)] proposed information erasure scheme, where the cost of erasure is solely in terms of a conserved quantity other than energy, allows for new kinds of heat engines. In recent work, we studied the discrete fluctuations and average bounds of the erasure cost in spin angular momentum. Here we clarify the costs in terms of the spin equ…
This work concerns the statistics of the Two-Time Measurement definition of heat variation in each reservoir of a thermodynamic quantum system. We study the cumulant generating function of the heat flows in the thermodynamic and large-time limits. It is well-known that, if the system is time-reversal invariant, this cumulant generating function satisfies the celebrated Evans--Searles symmetry. We show in addition that, under appropriate ultraviolet regularity assumptions on the local interaction between the reservoirs, it satisfies a translation-invariance property, as proposed in [Andrieux et…
We study dynamic cooling, where an externally driven two-level system is cooled via reservoir, a quantum system with initial canonical equilibrium state. We obtain explicitly the minimal possible temperature $T_{\rm min}>0$ reachable for the two-level system. The minimization goes over all unitary dynamic processes operating on the system and reservoir, and over the reservoir energy spectrum. The minimal work needed to reach $T_{\rm min}$ grows as $1/T_{\rm min}$. This work cost can be significantly reduced, though, if one is satisfied by temperatures slightly above $T_{\rm min}$. Our results …
Classical thermodynamics is built with the concept of equilibrium states. However, it is less clear how equilibrium thermodynamics emerges through the dynamics that follows the principle of quantum mechanics. In this paper, we develop a theory to study the exact nonequilibrium thermodynamics of quantum systems that is applicable to arbitrary small systems, even for single particle systems, in contact with a reservoir. We generalize the concept of temperature into nonequilibrium regime that depends on the detailed dynamics of quantum states. When we apply the theory to the cavity system and the…
I formulate a quantum stochastic thermodynamics for the quantum trajectories of a continuously-monitored forced harmonic oscillator coupled to a thermal reservoir. Consistent trajectory-dependent definitions are introduced for work, heat, and entropy, through engineering the thermal reservoir from a sequence of two-level systems. Within this formalism the connection between irreversibility and entropy production is analyzed and confirmed by proving a detailed fluctuation theorem for quantum trajectories. Finally, possible experimental verifications are discussed.
So far, feedback-driven systems have been discussed using (i) measurement and control, (ii) a tape interacting with a system or (iii) by identifying an implicit Maxwell demon in steady state transport. We derive the corresponding second laws from one master fluctuation theorem and discuss their relationship. In particular, we show that both the entropy production involving mutual information between system and controller and the one involving a Shannon entropy difference of an information reservoir like a tape carry an extra term different from the usual current times affinity. We thus general…
We consider the global thermal state of classical and quantum harmonic oscillators that interact with a reservoir. Ohmic damping of the oscillator can be exactly treated with a 1D scalar field reservoir, whereas general non-Ohmic damping is conveniently treated with a continuum reservoir of harmonic oscillators. Using the diagonalized Hamiltonian of the total system, we calculate a number of thermodynamic quantities for the damped oscillator: the mean force internal energy, mean force free energy, and another internal energy based on the free-oscillator Hamiltonian. The classical mean force en…
Harnessing Measurement Error as a Thermodynamic Resource in Two-Reservoir Information Engines
College of Physics, Jilin University（吉林大学物理学院）；State Key Laboratory of Electroanalytical Chemistry, Changchun Institute of Applied Chemistry（中国科学院长春应用化学研究所电分析化学国家重点实验室）；Department of Chemistry and Department of Physics and Astronomy, State University of New York at Stony Brook（纽约州立大学石溪分校化学系和物理与天文学系）
Thermal machines perform useful tasks--such as producing work, cooling, or heating--by exchanging energy, and possibly additional conserved quantities such as particles, with reservoirs. Here we consider thermal machines that perform more than one useful task simultaneously, terming these "hybrid thermal machines". We outline their restrictions imposed by the laws of thermodynamics and we quantify their performance in terms of efficiencies. To illustrate their full potential, reservoirs that feature multiple conserved quantities, described by generalized Gibbs ensembles, are considered. A mini…
In conventional molecular simulation, metastable structures often survive over considerable computational time, resulting in difficulties in simulating equilibrium states. In order to overcome this difficulty, here we propose a newly devised method, molecular Monte Carlo simulation of systems connected to three reservoirs: chemical potential, pressure, and temperature. Gibbs-Duhem equation thermodynamically limits the number of reservoirs to 2 for single component systems. However, in conventional simulations utilizing 2 or fewer reservoirs, the system tends to be trapped in metastable states.…
We investigate the thermodynamics of simple (non-interacting) transport models beyond the scope of weak coupling. For a single fermionic or bosonic level -- tunnel-coupled to two reservoirs -- exact expressions for the stationary matter and energy current are derived from the solutions of the Heisenberg equations of motion. The positivity of the steady-state entropy production rate is demonstrated explicitly. Finally, for a configuration in which particles are pumped upwards in chemical potential by a downward temperature gradient, we demonstrate that the thermodynamic efficiency of this proce…
We consider a classical and possibly driven composite system $X \otimes Y$ weakly coupled to a Markovian thermal reservoir $R$ so that an unambiguous stochastic thermodynamics ensues for $X \otimes Y$. This setup can be equivalently seen as a system $X$ strongly coupled to a non-Markovian reservoir $Y \otimes R$. We demonstrate that only in the limit where the dynamics of $Y$ is much faster then $X$, our unambiguous expressions for thermodynamic quantities such as heat, entropy or internal energy, are equivalent to the strong coupling expressions recently obtained in the literature using the H…
In the current research, we investigate the concept of spontaneously nonequilibrium dimension (SND), and show that a SND-based system can break the second law of thermodynamics. The main characteristic of the SND is the inherent nonequilibrium particle crossing ratio. A locally nonchaotic energy barrier is employed to form the model system. On the one hand, when the barrier width is much smaller than the mean free path of the particles, the system cannot reach thermodynamic equilibrium; on the other hand, the nonequilibrium particle distribution allows for production of useful work in a cycle …
This paper has been withdrawn by the author. For the reason, see the bottom paragraph of this abstract. By generalizing Tasaki's work on the second law of thermodynamics for an adiabatic process between two equilibrium states of a macroscopic quantum compound system, we obtain an extension of the second law to a transient adiabatic process that takes a macroscopic quantum compound system consisting of a system of interest and two heat reservoirs from an initial equilibrium state to a final non-equilibrium state, where the system of interest is in a steady heat conduction state. For the system …
The development of efficient artificial nanodevices poses challenges which are of fundamental and technological nature. Recent progress has been made in the context of finite-time thermodynamics. A central question in finite-time thermodynamics is to identify the optimal procedure to extract the greatest amount of work from a system operating under well-defined constraints. For externally controlled small systems, the optimal driving protocol maximizes the mean work spend in a finite-time transition between two given system states. For simplicity we consider an externally controlled single lev…
We study the modification of the second law of thermodynamics for a quantum system interacting with a reservoir regarding quantum coherence. The whole system is isolated so that neither energy nor information is lost. It is discovered that the coherence of the reservoir can serves as a useful resource allowing the system extract more energy from the reservoir; among the coherence measures, only is the relative entropy of coherence feasible to quantitatively characterize energy exchange. We demonstrate that a thermodynamic cycle between two coherent reservoirs can output more work than its clas…
We study the coarse-graining approach to derive a generator for the evolution of an open quantum system over a finite time interval. The approach does not require a secular approximation but nevertheless generally leads to a Lindblad-Gorini-Kossakowski-Sudarshan generator. By combining the formalism with Full Counting Statistics, we can demonstrate a consistent thermodynamic framework, once the switching work required for the coupling and decoupling with the reservoir is included. Particularly, we can write the second law in standard form, with the only difference that heat currents must be de…
